contour integral

Let $f$ be a complex-valued function defined on the image of a curve (http://planetmath.org/Curve) $\alpha$: $[a,b]\to ℂ$, let $P=\{a_0,\mathrm{\dots },a_n\}$ be a partition (http://planetmath.org/Partition3) of $[a,b]$. We will restrict our attention to contours, i.e. curves for which the parametric equations consist of a finite number of continuously differentiable arcs. If the sum

$$\sum _{i=1}^{n}f(z_i)(\alpha (a_i)-\alpha (a_{i-1})),$$

where $z_i$ is some point $\alpha (t_i)$ such that $a_{i-1}⩽t_i⩽a_i$, converges as $n$ tends to infinity and the greatest of the numbers $a_i-a_{i-1}$ tends to zero, then we define the contour integral of $f$ along $\alpha$ to be the integral

$${\int }_{\alpha }f(z)𝑑z:={\int }_a^{b}f(\alpha (t))𝑑\alpha (t)$$

Notes

(i) If $\mathrm{Im}(\alpha )$ is a segment of the real axis, then this definition reduces to that of the Riemann integral of $f(x)$ between $\alpha (a)$ and $\alpha (b)$.

(ii) An alternative definition, making use of the Riemann-Stieltjes integral, is based on the fact that the definition of this can be extended without any other changes in the wording to cover the cases where $f$ and $\alpha$ are complex-valued functions.

Now let $\alpha$ be any curve $[a,b]\to {ℝ}^2$. Then $\alpha$ can be expressed in terms of the components $({\alpha }_1,{\alpha }_2)$ and can be associated with the complex-valued function

$$z(t)={\alpha }_1(t)+i{\alpha }_2(t).$$

Given any complex-valued function of a complex variable, $f$ say, defined on $\mathrm{Im}(\alpha )$ we define the contour integral of $f$ along $\alpha$, denoted by

$${\int }_{\alpha }f(z)𝑑z$$

by

$${\int }_{\alpha }f(z)𝑑z={\int }_a^{b}f(z(t))𝑑z(t)$$

whenever the complex Riemann-Stieltjes integral on the right exists.

(iii) Reversing the direction of the curve changes the sign of the integral.

(iv) The contour integral always exists if $\alpha$ is rectifiable and $f$ is continuous.

(v) If $\alpha$ is piecewise smooth and the contour integral of $f$ along $\alpha$ exists, then

$${\int }_{\alpha }f𝑑z={\int }_a^{b}f(z(t))z^{\prime }(t)𝑑t.$$

Title	contour integral
Canonical name	ContourIntegral
Date of creation	2013-03-22 12:51:44
Last modified on	2013-03-22 12:51:44
Owner	Mathprof (13753)
Last modified by	Mathprof (13753)
Numerical id	23
Author	Mathprof (13753)
Entry type	Definition
Classification	msc 30A99
Classification	msc 30E20
Synonym	complex integral
Synonym	line integral
Synonym	curve integral
Related topic	CauchyIntegralFormula
Related topic	PathIntegral
Related topic	Integral
Related topic	IntegralTransform
Related topic	RealAndImaginaryPartsOfContourIntegral
Defines	contour